Algebra universalis

, Volume 65, Issue 4, pp 353–362 | Cite as

On the finite basis problem for the monoids of triangular boolean matrices



Let \({{\fancyscript{T}\fancyscript{B}_n}}\) denote the submonoid of all upper triangular boolean n × n matrices. It was shown by Volkov and Goldberg that \({{\fancyscript{T}\fancyscript{B}_n}}\) is nonfinitely based if n > 3, but the cases when n = 2, 3 remained open. In this paper, it is shown that the monoid \({{\fancyscript{T}\fancyscript{B}_2}}\) is finitely based, and a finite identity basis for the monoid \({{\fancyscript{T}\fancyscript{B}_2}}\) is given. Moreover, it is shown that \({{\fancyscript{T}\fancyscript{B}_3}}\) is inherently nonfinitely based. Hence, \({{\fancyscript{T}\fancyscript{B}_n}}\) is finitely based if and only if n ≤ 2.

2010 Mathematics Subject Classification

Primary: 20M07 Secondary: 08B05 

Keywords and phrases

Finite basis problem semigroup of triangular matrices finite field semigroup variety finite semigroup 


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsLanzhou UniversityLanzhouP. R. of China

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